It is also possible to build approximations or alternative distances on top of OT computations. An option is to integrate OT distance along 1-D projection, which corresponds to the so-called sliced-Wasserstein distance [1]}.
Combining 1-D OT maps along the axes defines the Knothe map [2]}, which is a limit of OT maps for degenerate cost functions, and is related to the subspace detours variation of OT [3]}.
One can also approximate OT between mixtures of Gaussians by leveraging the closed form expression of OT between Gaussians [4]}, which finds applications in texture synthesis [5]}.
